Page 242 - Applied Probability

10. Molecular Phylogeny
228
Use the representation (10.18) of the partition function to prove that

m
H(c)
m
= φ + φ ,
e
2
1
c
where φ 1 and φ 2 are the eigenvalues of Z. What are these eigenvalues?
(Hints: Express the partition function as a matrix trace, and use the
t
identity uu = I.)
u
10.12 References
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[2] Cressie NAC (1993) Statistics for Spatial Data. Wiley, New York
[3] Derin H, Elliott H (1987) Modeling and segmentation of noisy and tex-
tured images using Gibbs Random Fields. IEEE Trans Pattern Anal
Mach Intelligence 9:39–55
[4] Eck RV, Dayhoff MO (1966) Atlas of Protein Sequence and Structure.
National Biomedical Research Foundation, Silver Spring, MD
[5] Felsenstein J (1978) Cases in which parsimony or compatibility meth-
ods will be positively misleading. Syst Zool 27:401–410
[6] Felsenstein J (1981) Evolutionary trees from DNA sequences: A max-
imum likelihood approach. J Mol Evol 17:368–376
[7] Felsenstein J, Churchill GA (1996) A hidden Markov model approach
to variation among sites in rate of evolution. Mol Biol Evol 13:93–104
[8] Fitch WM (1971) Toward defining the course of evolution: Minimum
change for a specific tree topology. Syst Zool 20:406–416
[9] Goldman N, Yang Z (1994) A codon-based model of nucleotide substi-
tution for protein-coding DNA sequences. Mol Biol Evol 11:725–736
[10] Hartigan JA (1973) Minimum mutation fits to a given tree. Biometrics
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[11] Hirsch MW, Smale S (1974) Differential Equations, Dynamical Sys-
tems, and Linear Algebra. Academic Press, New York
[12] Kelly FP (1979) Reversibility and Stochastic Networks. Wiley, New
York
[13] Kimura M (1980) A simple method for estimating evolutionary rates
of base substitutions through comparative studies of nucleotide se-
quences. J Mol Evol 16:111–120